Figure[White to move]

Black’s rook is loose, but there is no way for White’s queen to attack it and give check at the same time. So look at the checks White can give and consider their consequences. There are five such possibilities: Qa8, Qe8, Qh7, Qh4, and Qh1. A couple of them—Qa8 and Qh7—can be dismissed without any real thought. But then there are the others: (a) Qe8+ forces the king to h7; this almost works because it then allows Qe7+, attacking king and rook. But of course Black replies RxQ. (b) Qh4+ forces Kg8 but leaves White with no good follow-up; from h4 there is no way to attack king and rook simultaneously. (c) Qh1+ also forces Kg8. But this time White does then have a move that attacks king and rook at the same time: Qg1+, winning the rook.